Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,\lambda )$-graphs and symmetric weighing matrices

In this paper, we illustrate important aspects of the interplay between weighing matrices, $(v,k,\lambda )$-graphs with fixed-point free involutions, and signed graphs with an orthogonal adjacency matrix, which arises from thin divisible design graphs. In particular, we present two new recursive constructions of regular symmetric Hadamard matrices with constant diagonal (equivalently, two new recursive constructions of strongly regular graphs) and we find a fixed-point free involution in the symplectic graph $Sp(4,q)$, where $q$ is odd, which leads to orthogonal signings for an infinite family of antipodal distance-regular graphs of diameter 3.